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Why Directional Mics are Inherently Colored
There are a substantial number of microphones with an essentially flat on-axis frequency response (to within about one or two dB), but without exception these mics are omnidirectional. Typically these mics are designed for making acoustical measurements, not music recordings, but can work well for both applications. Notable brands that sell these types of mics are Bruel & Kjaer, DPA, and Earthworks.
Nonetheless, for music recording an omnidirectional mic is not always the best option, but switching to a directional microphone, such as cardioid, will inevitably result in a different, and not so flat, frequency response. Of course, much of the reason that people love certain microphones is for their colored frequency response, but this is very application specific and in some cases flat might be the best option.
Part of the reason for this non-flat response of directional microphones is due to proximity effect. All directional (i.e. non-omni) microphones have a change in bass response that increases the closer the sound source is to the mic. At best a directional microphone can be flat at a single, usually unspecified, distance. When a source is closer than this distance the low-frequency response will be boosted and when the source is farther than this distance a loss in bass will occur.
Some directional mics can partially compensate for proximity effect, such as the Electro-Voice RE-20 (and its offspring), but the operative word is partially. While in theory, it’s possible to create a directional microphone with no proximity effect, in reality, it’s just not practical for a multitude of reasons.

The diagrams above show the fundamental differences in the design between an omnidirectional capsule and a figure-eight capsule. A cardioid capsule can be thought of as a combination of these two designs, and therefore is inherently more complex.
As you can see, dedicated single-pattern omnidirectional mics are effectively just a diaphragm mounted in a sealed capsule, which tend to be flatter, in part, due to this simpler design. On the other hand, to be directional the capsule must have rear sound ports with acoustical phase shift networks, which for any practical design has inherent non-idealities. One of the flattest directional mics at high frequencies is the Sennheiser MHK30 figure-8 microphone, although being figure-8 means that it is especially sensitive to proximity effect and at somewhat close distances is far from flat at low frequencies.
Some Like it Flat
While the Sphere microphone is designed to model other microphones, it can also create new kinds of microphones that don’t exist in reality. One of these new types is the “Sphere Linear” microphone model, located under the “Custom” mic type category.
Just like any large diaphragm condenser, the frequency response of the Sphere microphone is not perfectly flat, but it is possible to apply DSP processing with a compensation filter so you end up with a nearly flat response. The Sphere Linear model by default is flat at a distance of about 50cm, but by enabling Off-Axis Correction™ it's possible to have a flat response any distance. Off-Axis Correction™ has an On-Axis Distance control and an Off-Axis Distance control. When using the Sphere Linear model the On-Axis Distance control adjusts at what distance the on-axis frequency response will be flat. The Off-Axis Distance control adjusts at what distance the polar pattern will have the most off-axis rejection.
Of course, it’s not unusual to record multiple sources with one mic, such as an orchestra. This might result in only some instruments having a flat response, although once the sources are more than about four meters the distance doesn’t matter that much. Like a zoom camera lens, you can set the focal point to infinity as long as the sources are far enough away.
Flat Off-Axis
Sphere also goes a significant step further by flattening the off-axis response. Although it’s impossible to perfectly flatten the response simultaneously in all directions, it is possible to produce a considerably more ideal polar response.
Another Custom microphone model is "Sphere Linear Diffuse" which provides a flatter response for sound coming from all directions, such as reverberation. This can be particularly useful for room miking or other techniques where the majority of sound is off-axis.
If desired, it's even possible to adjust the Sphere Linear model so that it's flat in a particular off-axis direction. This can be useful in live situations where the microphone cannot be placed on-axis because it will block the audience's view of the musician, or because the microphone is handheld and the most comfortable position for the vocalist is somewhat off-axis. This can also be useful when using a single Sphere microphone to record in stereo because the dominant sound sources are likely to be off-axis.
For example, if the microphone is 45 degrees off-axis to the dominant source then setting the Axis control to 45 degrees will make the microphone flat at 45 degrees off-axis. Keep in mind this does mean that the microphone will now not be as flat on-axis. Depending on the pattern this axis correction works up to about 90 degrees, and best results are obtained in the +/-45 degree range. Beyond 90 degrees the Axis control provides a moderate high-frequency roll off.
Multi-pattern large diaphragm condenser microphones tend to sound especially colored when in figure-eight mode, and usually not in a good way. The high-frequency response is far from flat and proximity effect can be very strong, so it can be hard to get good results. Hyper-cardioid settings can also be problematic, but generally to a lesser degree. Figure-eight ribbons mics have a much smoother high-frequency response, but they also have substantial high roll-off and large proximity effect that isn’t necessarily desirable.
With Sphere you can make a nearly flat figure-eight mic, which in many use cases is just not possible with any other type of microphone. This can be especially useful if you need the excellent side rejection that a figure-eight mic can provide, but without the coloration.
Ocean Way Studios Plug-In
Another interesting use case is with the Ocean Way Studios plug-in from Universal Audio.  When set to Re-Mic mode this plug-in models a collection of prized vintage microphones combined with the acoustics of the studio. To get results that are as close as possible to actually recording there, it is advisable to feed the plug-in a dry and neutral signal, so that the modeling has a good starting point to work from.
Any coloration in the original source mic will, of course, color the modeled result. Likewise, reflections from the room that the source audio was recorded in will also color the result. With this in mind, we recommend using the Sphere Linear model with the Off-Correction Pattern set to hypercardioid to get the least amount of room pickup.
Is Flatter Better?
Most recording engineers are not used to hearing flat microphones, and flat might sound boring or maybe even slightly odd. Even though it might theoretically be more accurate, in the end, it’s a subjective artistic choice depending on many factors such as the musical genre, the creative vision, and the sound sources themselves.
But you can also use a flat response that the Sphere technology provides as a starting point for applying EQ to get that sound you’re looking for. Often just some low and high shelves with a small amount of boost, or a modest tweak to the Proximity EQ control, can do the trick. But of course, you can also create any microphone signal chain you envision by adding third-party plugins, such as EQ and compression, while maintaining full control of the polar pattern.